Problem: Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{q^2 + 5q + 6}{q^2 + 2q}$
First factor the expressions in the numerator and denominator. $ \dfrac{q^2 + 5q + 6}{q^2 + 2q} = \dfrac{(q + 3)(q + 2)}{(q)(q + 2)} $ Notice that the term $(q + 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q + 2)$ gives: $t = \dfrac{q + 3}{q}$ Since we divided by $(q + 2)$, $q \neq -2$. $t = \dfrac{q + 3}{q}; \space q \neq -2$